The $100 Switch-A-Roo: Answer to Brainteaser

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At the end of the program “What Were You Thinking,” Professor Debbie Bennett left listeners with a brain teaser, the answer supplied here.

Dr. Bennett writes: “The problem involved two identical envelopes–one with two $1 bills in it and the other with a $1 bill and a $100 bill. The envelopes are mixed and you choose one to keep. But before you get to keep it, I randomly remove a bill from your envelope and show it to you. It is a $1 bill. I return it to your envelope and offer to switch envelopes with you. Do you keep your envelope or switch with me? In other words, is your envelope more likely, less likely, or equally-likely to contain the $100 bill?

ANSWER: You should switch. My envelope is twice as likely to contain the $100 bill. If your envelope has the two $1 bills, there is a 100% chance that when I remove a bill, it is a $1 bill. If your envelope has the $1 bill and the $100 bill, I have only a 50% chance of removing a $1 bill.

Since I did remove a $1 bill, it is more likely to have come from the envelope with the two $1 bills. Considered another way: Since you saw me remove a $1 bill, there are three possible scenarios:

1) You have the $1 and $100 and I removed the $1.

2) You have the two $1 bills and I removed the first one (the second remained hidden).

3) You have the two $1 bills and I removed the second one (the first remained hidden).

In 1 out of 3 equally-likely scenarios, you have the $100 bill and in 2 out of the 3 scenarios, I have the $100.”

Any questions? Send a $100 to “Are We Alone?” and we’ll answer it for you (or, leave a comment and Dr. Bennett will)!